4. Extract the square cul>e(i> or §ixth root, of 43572838 

 1009267809889764416. 



Ans. 27534, 



5. Find the seventh jroot of 344B771 74673075 13 18249 

 ^153794673, 



Ans. 32017. 



6. Find the eighth root of 11210162813204762362464 

 97942460481. 



Ans. 13527. 



to extract any root whatever by 

 Approximation. 



RULE. 



1. Assume the root nearly, and raise it to the same 

 power with the ^iven number, which call the assumed 

 power, 



2. Then, as the sum of the assumed power multiplied 

 by the index more i and the given number muUiplied by 

 the index less i, is to the sum of the given number multi- 

 plied by the index more i and the assumed power mul- 

 tiplied by the index less i, so is the assumed root to the 

 required root. 



Or, as half the first sum is to the difference between the 

 given and assumed powers, so is the assumed root to the 

 difference between the true and assumed roots ; which 

 difference, added or subtracted, gives the true root nearly. 



And the operation may be repeated as often as we please, 

 by using always the last found root for the assumed root, 

 and its power as aforesaid for the assumed power. 



EXAMPLES. 



I. Required the fifth root of 21035*8. 



Here 



